water ingestion into axial flow compressors · 2011-05-14 · 2.2 compressibility of the mixture 9...

AFAPL-TR-76-77

WATER INGESTION INTO AXIAL FLOWCOMPRESSORS

PURDUE UNIVERSITYSCHOOL OF AERONAUTICS AND ASTRONA UTICS

S WEST LAFAYETTE, INDIANA 47907

AUGUST 1976

TECHNICAL REPORT AFAPL-TR-76-77FINAL REPORT FOR PERIOD 1 AUGUST 1975 - 31 AUGUST 1976 - "'.

Approved for public release; distribution unlimited

AIR FORCE AERO PROPULSION LABORATORYAIR FORCE WRIGHT AERONAUTICAL LABORATORIESAIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AIR FORCE BASE, OHIO 45433

IM\

NOTICE

Whet, Government drawings, specifications, or other data are usedfor any purpose other than in connection with a definitely relatedGovernment procurement operation, the United States Government therebyincurs no responsibility nor any obligation whatsoever; and the factthat the Government may have formulated, furnished, or in any waysupplied the said drawings, specifications, or other data, is not tobe regarded by implication or otherwise as in any manner licensing theholder or any other person or corporation, or conveying any rightsor permission to manufacture, use, or sell any patented inventionthat may in any way be related thereto.

This final report was submitted by Purdue University undercontract F33615-74-C-2014. The effort was sponsored by theAir Force Aero Propulsion Laboratory, Air Force Systems Command,Wright-Patterson AFB, Ohio under Project 3145, Task 314532 and Work Unit31453220 with James R. HcCoy/DOY as project engineer in-charge.Dr. S.N.B. Murthy of Purdue University was technically responsiblefor the work.

This report has been reviewed by the Information Office, (ASD/OIP)and is releasable to the National Technical Information Service (NTIS).At NTIS, it will be available to the general public including foreignnations.

This technical report has been reviewed and is approved for publication.

AMES WILLIAM A. BUZZELL, I/LTSAFd Project Engineer Task Engineer

FOR THE COMMANDER

EO S.-HAROOTY , .

Chief, Techn4 tiviti s

4 Copies of this report should not be returned unless return is requiredby security considerations, contractual obligations, or notice on aspecific document.A O - t

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Final i-7 0Water Ingestion Into Axial Flow Compressorse 1 Auq 75 -: 31 Au0 a6

114o' H-WPAFB-T-76-l:P."CO TACT-o0 GA ii U-Yg DER($)

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19. KEY WO1OS (Continue on reverse aide It necessary and Idenify by block number)

Water ingestion, turbomachinery, and jet engines.

20 ABSTRACT (Contlinue on tov.ras side Hi necessary and Identify ,y block number)The problem of the flow of a gas-liquid mixture through a multi-stage axialcompressor originally designed for air flow arises during take-off from arough runway with water on it and during rairi. Preliminary investigationshave revealed the problem areas in the fan, L.P., and H.P. compressor stages.The basic aerothermodynamic equations have been deduced in a form suitablefor considering development of a computational program.

DD 1JAN 73 1473 EDITION OF I NOV 65 IS OVSOLeTT Unclassified ?Z 47SECURITY CLASSIFICATION OF TNJIS PAGE (When Data Entr

TABLE OF CONTENTS

PageABSTRACT ii

LIST OF FIGURES iii

LIST OF TABLES v

NOMENCLATURE vi

1. INTRODUCTION 11.1 Outline of Report 5

2. TWO PHASE FLOW CONSIDERATIONS 62.1 Water-Phase Continuum 82.2 Compressibility of the Mixture 92.3 Distribution of Droplet Size and Number Density 92.4 Break-up of Droplets 102.5 Formation of a Film of Water 112.6 Drag on a Cloud of Droplets 112.7 Heat (Mass) Transfer Between a Cloud of Droplets and

the Surrounding Fluid 14

3. PERFORMANCE ESTIMATION 203.1 Single Stage Compressor 213.2 Multistage Compressor: N.G.T.E. #109 213.3 P&W TF-30 Compressor 22

4. COMPRESSOR AEROTHERMODYNAMICS 234.1 General Considerations 234.2 Conservation Equations 25

5. DISCUSSION 305.1 Recommendations 32

6. REFERENCES 34

FIGURES 40

APPENDIX I 73

APPENDIX II 76II.1 Three-Dimensional Flow Equations in Intrinsic Coordinates 7611.2 Axisymmetric Flow Equations in Intrinsic Coordinates 8811.3 Three-Dimensional Flow Equations in Cylindrical

Coordinates 9311.4 Axisymmetric Flow Equations in Cylindrical Coordinates 104

(i)

ABSTRACT

The problem of the flow of a gas-liquid mixture through a multi-

stage axial compressor originally designed for air flow arises during

take-off from a rough runway with water on it and during rain. The

problem is complicated because of the presence of discrete droplets in a

rotating compressor flow field and because of the phase change during

compression. In order to alleviate this problem, it is necessary to

examine both changes in blade and casing design as well as possible

additional controls including variable geometry. The objectives of the

investigation presented here are (a) to establish estimates of possible

changes in performance of typical aircraft compressors operating with

two-phase flow, (b) to determine the governing aerothermodynamic equations

for two-phase flow in compressors, and (c) to outline directions in which

further research tay prove fruitful in this subject. Preliminary

investigations have revealed the problem areas in the fan, L.P., and H.P.

compressor stages. The basic aerothermodynamic equations have been

deduced in a form suitable for considering development of a computational

program.

tI

(ii)

LIST OF FIGURES

4 FigureVelocity of Sound vs. Liquid Volume Fraction

(At Standard Atmospheric Conditions) 2.1

Purdue 001 Compressor - Constant Reaction

100% Air at 429.04"K - 95% Air + 5% Steam at 3730K -

Blade Tip Values

Pressure Ratio vs. Mass Flow (RPM) 3.1a

Efficiency vs. Mass Flow (RPM) 3.1b

Purdue 002 Compressor - Free Vortex

100% Air at 429.040K - 95% Air + 5% Steam at 373°K -

Blade Tip Values



Purdue 001 Compressor - Constant Reaction

100% Air at 3730K and 2880K - 95% Air + 5% Steam at 3730K -

Blade Tip Values


Efficiency vs. Mass Flow (RPM)

100% Air at 288°K 3.3b

100% Air at 3730K - 95% Air + 5% Steam at 3730K 3.3c


100% Air at 373°K and 2880K - 95% Air + 5% Steam at 3730K -

Blade Tip Values


Efficiency vs. Mass Flow (RPM)

100% Air at 2880K 3.4b

100% Air at 3730K - 95% Air + 5% Steam at 3730K 3.4c

(iii)

Figure

- Purdue 001 Compressor - Constant Reaction

100% Air at 373 0K - 95% Air + 5% Steam at 3730K - Blade Tip

Values

Pressure Ratio vs. Mass Flow (Corrected RPM) 3.5a

Efficiency vs. Mass Flow (Corrected RPM) 3.5b


100% Air at 373°K - 95% Air + 5% Steam at 373°K - Blade Tip

Values



Purdue 001 Compressor - Constant Reaction and Purdue 002 - Free

Vortex

100% Air at 373 0K - Blade Mean Values

Pressure Ratio vs. M-ss Flow (RPM) 3.7a


Purdue 001 Compressor - Constant Reaction and Purdue 002 - Free

Vortex

100% Air at 373°K - Blade Mean Values



Purdue 001 Compressor - Constant Reaction and Purdue 002

Compressor - Free Vortex

Total Head Loss and Deflection vs. Incidence 3.9APurdue 001 Compressor - Constant Reaction

Air Angle vs. Radius Ratio

Design Conditions 3.10a

100% Steam at 373°K 3.10b

(iv)

FigurePurdue 002 Compressor - Free Vortex

Air Angle vs. Radius Ratio

Design Conditions 3.11a

100% Steam at 3730K 3.11b



Inlet Whirl Velocity vs. Blade Height 3.12



Exit Whirl Velocity vs. Blade Height 3.13

N.G.T.E. #109 Compressor

100% Air at 484.79°K - 90% Air + 10% Stean at 3730K -

Carter's Data

Pressure Ratio vs. Mass Flow 3.14a

Temperature Ratio vs. Mass Flow 3.14b

Stage Efficiency vs. Mass Flow

9500 RPM 3.14c

8000 RPM - 6000 RPM 3.14d

Intrinsic Coordinate System 4.1

Cylindrical Coordinate System 4.2

Dynamics of Wheel Spray A.I.l

ALIST OF TABLES

Table

Local Values of W s Under Saturated Conditions 1.1

(v)

NOMENCLATURE

A - droplet (cloud) surface area

Ap - projected frontal area of an equivalent sphere having

the same mass as distorted droplet

Aps - projected frontal area of a droplet having statistical

mean diameter, d s

CD - droplet drag coefficient

Cwg - water vapor concentration in gas surrounding droplet

(cloud)

C ws - water vapor concentration at surface of droplet (cloud)

c - mixture local sonic velocity

ci - constant 1

D - drag on a single droplet in a flowing gas

Db - binary mass diffusivity

D(a) - drag coefficient which depends on size of ds particle

and statistical properties of Ns particles

d - individual droplet diameter

dqg - heat loss from gas phase (per unit volume of mixture)

to surroundings, excluding heat lost directly to liquid

phase (joules/Vol-)

dqI - heat loss from gas phase (per unit volume of mixture)

directly to liquid phase (joules/Vol-')

(vi)

W •i

dqq - heat loss from liquid phase (per unit volume of mixture)

to surroundings, excluding heat lost directly to gas

phase (joules/Vol-)

ds - statistical mean diameter of a cloud of liquid droplets

(Sauter mean diameter)

ds g - rise in entropy of gas phase (per unit volume of

mixture) due to action of body and surface forces on

gas phase, excluding those forces imposed by liquid

phase (joules/*K Vol-)

dsI - rise in entropy of gas phase (per unit volume of the

mixture) due to action of forces on gas phase imposed

by liquid phase (joules/°K Vol-)

ds q - rise in entropy of liquid phase (per unit volume of the

mixture) due to action of body and surface forces on

liquid phase, excluding those forces imposed by gas

phase (joules/°K Vol])

FBgt - summation of t-components of all body forces exerted on

gas phase per unit volume of mixture (nwt/Volj )

F Bpi: - summation of t-compcnents of all body forces exerted on

liquid phase per unit volume of mixture (nwt/Vol-)

Flt - summation of t-components of all forces exerted on gasphase (per unit volume of mixture) imposed by liquid

phase (nwt/Vol-)

(vii)

FVgt - summation of t-components of all surface forces exertedon gas phase (per unit volume of mixture), excluding

those forces exerted by liquid phase (nwt/Vol-)

FVp t - summation of t-components of all surface forces exerted

on liquid phase (per unit volume of mixture), excluding

those forces exerted by gas phase (nwt/Vol-)

F(1,p) - defined by equation (2.28)

f - friction coefficient

g - acceleration due to gravity

Hg - total enthalpy of gas phase per unit mass of the gas

phase (joules/Kg )

H.P. - high pressure

h - standing water depth

h - enthalpy of gas phase per unit mass of gas phase

(joules/Kg)

hh - average interfacial heat transfer coefficient of

droplet (cloud)

hm - average interfacial mass transfer coefficient of

droplet (cloud)

Vi g 2 2- rothalpy (I = h + _ _ U

k - thermal conductivity of fluid surrounding droplet

(cloud)

(viii)

ke - the effective thermal conductivity of mixture

L - direction tangent to computing station at a particular

streamline (Figure 4.1)

L.P. - low pressure

L.P.C. - low pressure compressor

m - mass

- mass flow rate

m(n) - the nth moment of droplet distribution function

m,o,n - intrinsic coordinate system defined in Figure 4.1

v - water vapor mass production rate per unit volume of

mixture (gm/sec Vol5)

m.w. - molecular weight

mw - the amount of water sprayed (per unit distance of wheel

travel) behind the wheel

- time rate of the "pumping" of water by tire

Ns - statistical droplet number density of a cloud of liquid

droplets with statistical mean diameter, ds

Nu - Nusselt number

n(ar) - droplet distribution functionIr

nwt - Newtons

P - pressure

(ix)

0

-i4ai -:

P!-~ shaft work (per unit time per unit volume of mixture)

done on phase designated, P or9 p

Peh - heat transfer Peclet number

Pe M - mass transfer Peclet number

Po - total pressure

PO /POl - compressor total pressure ratio

P.- - Prandtl number

p(d) - probability of droplet having diameter between d and

(d + 6d)

q'gt - heat transferred from gas phase (t-direction), exclud-

ing heat transferred directly to liquid phase

(watts/meters )

qI it - heat transferred from gas phase (t-direction) to liquid

phase directly (watts/metersi)

qI pt - heat transferred from liquid phase (t-direction) to

surroundings, excluding heat transferred directly to

2gas phase (watts/meters.)

Rep - particle Reynolds number

R - gas phase gas constant

RPM(rpm) - revolutions per minute of compressor rotor

r - compressor radial coordinate defined in Figures 4.1

and 4.2

(x)

rc - stream surface meridional radius of curvature

r,o,z - cylindrical coordinate system defined in Figure 4.2

Sc - Schmidt number

Sh - Sherwood number

T - temperature

TgP - mass averaged gas phase-liquid phase temperature (OK)

To - total temperature

To 2/To - compressor total temperature ratio

U - magnitude of local blade rotational velocity vector

U - local blade rotational velocity vector

Uc - speed of undercarriage of an aircraft wing

Uoe x - mean fluid velocity prior to addition of ds size test

particle

Up - total internal energy of liquid phase (joules/Kg )

uE - relative superficial velocity (I - aV)

U P- internal energy of liquid phase per unit mass of liquid

phase (joules/Kgp )

V - magnitude of the absolute velocity vector

V - absolute velocity vector

W - magnitude of the relative velocity vector

- relative velocity vector

(xi)

*1 We - Weber number

W - saturation mass fraction of water vapor at given values

of temperature and pressure

X - defined by equation (2.26)

Y - defined by equation (2.25)

YB - defined in Figure A.I.1

YB - time derivative of y,

Z - defined by equation (2.27)

- thermal diffusivity

a - defined by equation (2.12)

al - rotor absolute flow inlet angle

a2 - rotor absolute flow exit angle

a - defined by equation (2.22)

01 - rotor relative flow inlet angle

a2 - rotor relative flow exit angle

Y - specific heat ratio

Yc - defined in Figure 4.1

S- blade row mean flow deflection

n- compressor efficiency

nl - defined by equation (A.I.3)

(xii)

Sn2 - defined by equation (A.I.4)I

- absolute viscosity

- interfacial viscosity

p - density

a - droplet surface tension

am - mass of particulate liquid (liquid phase) per unit

mass of mixture

a r - individual droplet radius

- volume of particulate liquid (liquid phase) per unit

volume of mixture

0 9 - heat generated (per unit time per unit volume ofg

mixture) in gas phase due to viscous dissipation caused

by all sources except liquid phase (watts/Vol-)

- heat generated (per unit time per unit volume of

mixture) in gas phase due to viscous dissipation caused

by liquid phase (watts/Vol-)

(P P- heat generated (per unit time per unit volume of

mixture) in liquid phase due to viscous dissipation

caused by all sources except gas phase (watts/Vol.)

0 - defined in Figure 4.1

1 - angle between radius vector and m-direction

02 - angle between radius vector and n-direction

(xiii)

g; mgO

- angular velocity of compressor rotor

- blade row mean total pressure loss

( )a - pertaining to air

( ) - pertaining to gas phase (gas phase = air + water vapor)

( )- - pertaining to mixture (gas phase + liquid phase)

( )m,e,n; r,o,z - pertaining to m,e,n; r,e, or z direction respectively,

(e.g., Vgm is magnitude of gas phase velocity in m-

direction and Vpz is magnitude of liquid phase velocity

in z-direction)

( ) - pertaining to particulate liquid phase (particulate

liquid phase = liquid phase)

( )w - pertaining to water

Ii

(xiv)

1. INTRODUCTION

Two-phase flow in turbomachinery is of interest in a number of

technological probiems, e.g. rain ingestion into compressors on ground or

during flight, condensation in steam turbines, pumping of crude oil,

circulation of coolants in nuclear reactors, etc. The principal interest

in this Report is in the problem of air-water mixture flow through

diffusers and compressors. In gas turbines installed in aircraft, the

problem of water ingestion into the engine is known to lead to serious

corsequences, both by reducing the stall margin and occasionally leading

to after-burner blowout (Ref. 1). The ingestion of water may arise

during take-off from puddles on rough runways or during flight through a

rainfall. It is unclear at the moment if the problems of surge and blow-

out can be removed with simple bleed-type controls or if one has to

employ, for instance, casing treatment or even redesign of selected

blading in extreme cases. It is possible that no general solution can be

obtained for all types of compressors and al types of water ingestion,

but one may be able to establish general guidelines for design, control,

and operation of compressors. Such general guidelines may then be

adapted to suit given compressor configurations.

No adequate theory exists for the motion of a two-phase fluid through

a compressor. The compressor of interest in this Report is a rotating

machine, for e.g., an axial flow compressor. The increase in pressure

j arises on account of work done on the fluid. In the absence of rotation

and work input to the fluid, one can consider a simple diffuser with an

adverse pressure gradient. There is no adequate theory for two phase

flow even through such a diffuser

11

The two-phase flow of interest in this Report has two basic

complexities:

(a) in the air-water mixture flow, water is in the form of discrete

droplets which may undergo break-up or coalescence during

motion.

(b) When work is done on the fluid, water may undergo a phase change,

leading ultimately to the fluid becoming an air-steam mixture.

The aerothermodynamics of two-phase flow thus becomes quite complicated.

Considering an axial compressor, such as for e.g. the TF-30, several

additional complexities arise in the flow:

(a) Alternating rotating and .tationary blade rows.

(b) Division of the compresscr into the L.P. and H.P. sections.

(c) Flow entry geometry at the inlet to the compressor.

Taken together with the complexities of two-phase flow, the foregoing

presents a formidable problem.

At this time, there are no experimental studies of a suffizicntly

detailed nature to provide some guidance for theory and analysis. The

experimental studies again present , complicated problems on

account of the rotation in the fluid (both centrifugal and Coriolis

forces have to be included appropriately) and the presence of the dis-

crete phase. For example, the combined effects of centrifuging and

coalescence can lead to a film flow of water along the casing of a

compressor and it is important to establish the possibility of such a

film flow both from an analytical and a practical point of view.

2

U

An interesting aspect of the presence of water at entry to the

4 compressor is the ---se in humidity of air throughout the compressor.

A measure : ir . *,, .jitv is what may be termed the "Saturation Mass

Fraction of Wate- , .rined az the mass of water in a pound mass of air

at saturation under -,iven conditions of pressure and temperature. Thus,

if W is the satur "- masc fraction of water at given values of ambient

temperature ar:d pre. sure, one can write (Ref. 2)

W 0.6219 PS/(P - P S) (0.1)

whereMolecular Weight of Water VaporMolecular Weight of Drj Air

P = air-water vapor mixture pressure

P5 = partial vapor pressure of water at conditions of

saturation.

Ws, by definition, is a local parameter. If one considers standa.rd

atmospheric conditions aL entry to the P&W TF-30 c'?mpressor ano the local

conditions at the exit of stages 1, 2, 3, 4, 5, and 6, one can tabulate

the values of Ws at the local conditions as shown in Table 1.1. Thus,

at a pressure of 39.71 psia and temperature of 419.78°K, the value of Ws

is 0.9821 which means that the amount of water that can be held in 1 lb

of air under saturated conditions is 0.9821 lb.

One snould however not be misled by the possible values of Ws. In

actual practice, the attainment of equilibrium conditions with respect to

i humidity requires a finite length of time which in turn depends upon the

size, number and gecgraphical distributions -f droplets, and their convec-

tion velocity. There is no clear method of establishing the increase in

3

Table 1.1

Local Values Of W sUnder Satur'ated Conditions*

Stage TemperatureJ Pressure s

OK psia

Inlet 271.41 11.90 0.0040

1 302.68 16.97 0.0217

1 332," 22.59 005

L.P.C. Inlet 365.54 24.74 0.2778

4 379.39 32.04 0. 352

5 419.78 39.71 0.9821

6 43.9 46.77 j>1 Not DefinedJ

*Fan Data Ta'ken At Diameter Equal to Outer Diameter Of L.P.C. CaptureArea. L.P.C. Data Taken At Blade Tips.

4

humidity of the air in the presence of water droplets over a unit period

of time even in a simple duct flow. In a complicated flow such as that

in a compressor, the problem becomes practically insurmountable. However,

one has to allow for some change in the liquid water content along the

compressor on account of changes in the humidity of air.

The objective of this Report is three-fold: (1) To conduct a

performance estimation study on a selected compressor for the general

effects of two-phase flow in the compressor when it has been originally

designed for air flo4 only; (2) To establish the basic governing equations

for the aerotheraiodynamics of two-phase flow through a compressor; and

(3) To outline a program of theoretical and experimental studies that can

lead to an engineering solution to ?he problem of water ingestion into

installed engine compressors.

1.1 Outline of Report

In order to set the background for the problem of two-phase flow

through a compressor we present in Section 2 some basic two-phase flow

considerations. The presence of water in discrete phase introduces in a

compressor problems of heat and mass transfer and of phase change. We

therefore discuss thermo-physical and transport considerations.

in Section 3 some preliminary results are presented on the changes

in the overall performance of selected compressors when two-phase flow is

introduced into them. A general discussion on aerothermodynamics of two-

phase flow through a compressor is presented in Section 4.

Finally, an outline for future studies is developed in Section 5

with a special emphasis on experimental studies.

5d

I

42.2. TWO-PHASE FLOW CONSIDERATIONS

The subject of two-phase flow is of interest in many technologies

and several reviews of the subject are available (Refs. 3-7) although the

subject is extremely complicated and presents many unsolved problems. We

shall concentrate here on those aspects of two-phase flow that are

specialized to the current investigation.

The two-phase flow of interest in this investigation is air carrying

water droplets. The water droplets are the result of rain ingestion or

of water sprayed into the inlet during take-off from puddles on uneven

runways.

The presence of water alters the humidity of air at entry and in

fact throughout the compressor with changes in temperature and pressure

so long as water is present in the flow

The first consideration when water is present in discrete droplet

form is the size and number density distribution. If the water is

ingested on account of rain, the initial size and number density distribu-

tion can be related in a very approximate fashion to the rain droplet

size distribution (Refs. 8-9). When water is splashed into the compressor

inlet from runway puddles, one can make a preliminary estimate of the

amount of water that may be splashed by the wheel on account of the

relative motion between the water surface and the adjoining air flow

(see Appendix I). There is considerably greater ambiguity in the spray

I characteristics (Ref. 10). In addition to the uncertainties in the spray

envelope from the wheel, there is also the difficulty of determining the

influence of inlet geometry and flow on the actual ingestion process.

6

miI

The inlet flow itself is altered in many cases by the proximity of the

t ground. Until such time when better data become available on droplet

size and number characteristics, the only feasible approach is to assume

that distributions of statistical mean diameter and number density may be

treated as parameters at the inlet to the compressor.

Upon entry ihto the compressor, the size and number density

distributions of droplets alter continuously on account of the following.

(i) break-up and coagulation; and

(ii) evaporation.

The break-up and coagulation arise on account of the motion in a moving

and generally rotating environment in the presence of gravitational

action. The evaporation requires the droplets to become heated to the

boiling point at the local pressure and then to evaporate with the

absorption of latent heat. The evaporation is a strongly rate dependent

process and several characteristic length and time scales are involved.

At this stage the only feasible approach to take in such problems is to

include the break-up and evaporation processes in the changes in the

statistical mean diameter and number density locally along the streamlines.

The second consideration when water is present in discrete droplet

form is the distribution of velocities for the different droplets. We

assign a single mean velocity for each group of droplets with an assigned

mean statistical diameter. We are thus avoiding consideration of the

mtion of individual droplets in any group.

VThe third consideration arises from the fact that water droplets

introduce the drag of droplets into the momentum balance of the air flow.

The drag is based here, by assumption, on the Reynolds number calculated

7

with respect to the statistical mean diameter and the relative velocity

A of the droplets with respect to air flow velocity. The drag of N.

droplets is taken as Ns times the drag of one ds size particle.

The fourth and last consideration to be taken into account here is

the difference between the compressibility of the mixture and the

compressibility of air. The acoustic velocity in the mixture is

different from that in the air and the local Mach number of the mixture

at any location is therefore based on the acoustic velocity in the

mixture.

In view of the foregoing considerations, some tentative bases have

been selected to take into account the various processes. They are

elucidated in the following.

2.1 Water-Phase Continuum

The water content in the mixture is treated as a continuum in the

general equations of motion along with air and the mixture. The water

content in a unit volume at any location is assigned a value av by

volume. Continuity is thus assumed for (I - a v) of air and av of

water. It should be pointed out that av changes along the flow and

hence is a dependent variable in the flow equations.

In spite of the assumption of continuum for the water droplets, it

is assumed that the water droplets do not contribute to the local

pressure based on the reasoning that the droplets are not distributed

randomly enough.iIt may be pointed out here that for certain purposes, the droplets

are considered as discrete entities. Three processes for which it is

considered necessary to discretize the droplets are (1) the exchange of

8

iI

momentum between the air and the droplets, (2) the evaporation of the

i| droplets, and (3) the break-up and agglomeration of droplets.

2.2 Compressibility of the Mixture

The acoustic velocity in a droplet-laden air flow is given by the

following relation (Ref. 11).

c = (l - OV)Pg + av } cl + (2.1)Sg2 p

where

c = mixture sonic velocity

av = particulate liquid volume fraction = volume of particulate

liquid per unit volume of mixture

p = density

( ) = pertaining to gas phase

( )p = pertaining to particulate liquid phase.

Equation (2.1) is presented graphically in Figure 2.1 for standard

atmospheric conditions and for a range of av from 0.00 to 0.10; the

range over which this report is interested.

2.3 Distribution of Droplet Size and Number Density

The volume vv of water in a unit volume of the mixture at any

location is specified by the following relation.

v ds INs (2.2)

where

ds statistical mean droplet diameter of a cloud of liquid droplets

11s statistical droplet number density of a cloud of liquid

droplets with a statistical mean diameter, ds.

9

W '

1For our purposes the statistical mean droplet diameter is taken as

the Sauter mean diameter (Ref. 12)

d Jfdmax d3p(d)dd/IImax d2p(d)dd (2.3)

where

d = an individual droplet diameter

p(d) = the probability of a droplet having a diameter between d

and (d + 6d).

It should be noted that a droplet with diameter equal to the Sauter mean

diameter has the same surface area to vulume ratio as the entire spray.

It should be recognized that in addition to av' as and Ns also may

change along the flow depending upon droplet break-up and evaporation

processes. Thus, when av changes, there may also arise a change in d.

and then one can assign a value for Ns corresponding to the changed

values of av and ds-

At entry to the compressor, a radial and circumferential distribu-

tion of av and ds is specified, althou'Th under an axisymmetric assumption

the circumferential variation in distribution can be neglected.

2.4 Break-up of Droplets

A droplet is assumed to breakup when the Weber number b~comes of the

order of ten. The Weber number based on ds is defined by the relation

(Ref. 13):

We = p (Vg V )2 ds/ (2.4)

10

where

We = Weber number

V = velocity

a = droplet surface tension.

It may be noted again that a is not affected by the break-up

locally, although the change in ov from position to position is affected

by the droplet break-up.

2.5 Formation of a Film of Water

The combined effect of droplet agglomeration, centrifugal action and

gravitational action can lead to the formation of a stream of water at

the casing wall. There is no experimental evidence for this and it is a

very important question both from the point of view of fluid dynamics

and the possible arrangements required for the removal of water. It is

extremely difficult to establish the formation of a film of water on an

analytical basis.

We assume that when the value of av reaches a large enough value

ior the mixture to be treated as a bubbly mixture, there arises a film.

Such a value of av is Lstimated to be 0.40 for tap water and air and

0.90 for "pure" water and air (Ref. 14) at a pressure of one atmosphere.

2.6 Drag on a Cloud of Droplets

The drag on a spherical droplet carried in an airflow is a function

of its Reynolds number based on the droplet diameter (ds) and the droplet

velocity relative to that of air velocity in the vicinity of the droplet.

If we assume d. = 1.0 mm and a relative velocity of 10 m/sec., the

Reynolds number becomes 688 at standard atmospheric temperature and

1!

i'.

pressure. The drag on a single droplet may then be written as follows

(Ref. 15).

2D = 0.5 CD Pg(Vg - V p)Ap (2.5)

where

CD = 24.0 ReP Rep < 0.5

CD = 27.0 ReP -°084 0.5 < Rep < 70.0

CD = 0.414 Rep 0.1433 70.0 < Rep < 1.3 x 106 (2.6)

CD = 3.0 1.3 x 106 < Rep.

The particulate Reynolds number is defined as follows.

Rep = Pg d sVg - Vpl/1g (2.7)

where

CD = particle drag coefficient

Ap = projected frontal area of an equivalent sphere having the

same mass as the droplet.

The drag on a cloud of droplets becomes much more complicated to

estimate because of (a) the mutual interaction between the droplets and

(b) the arbitrariness of the direction of motion of different droplets

at different times in the vicinity of any one droplet. References 16-28

provide a short bibliography on this subject. Two examples of drag

correlations are given here, one due to Refs. 16 and 21, and the other

due to Ref. 22.

2.6.1 Soo's Drag Coefficient Using Ergun's Pressure Drop Equation

Drag of a cloud of N droplets = N x the drag of one ds size particle2

Ns x 0.5 CD pg(g - Vp) ApC (2.8)

12

where

A projected frontal area of a droplet having the statisticalps

mean diameter ds.

200Cl vUg 7 (2.9)CD :(I )'ds u E pg

+ 3(1 - V)(2.9)

where

0.08 a < 0.60 and 1.0 Rev S 3000.0

P absolute viscosity

uE = relative superficial velocity/(l - ar).

The relative superficial velocity is the relative gas flow, (V - V )

based upon the unobstructed flow area, i.e., the flow area when no

droplets are present.

2.6.2 Tam's Formula for the Drag Exerted on a Particle in a Cloud of

Spherical Particles

Drag on one ds size particle = D(a) Uo ^x (2.10)

where

U0 ex = the mean fluid velocity before the addition of the d. size

particle

D(u) = a drag coefficient which depends on the size of the ds

particle and the statistical properties of the Ns particles

D(o) 3n .g d U +-2--+ ( d2 (2.11)

. - 3o v6, m2 + [36n

2 m2 + 247r mI(l - 2)] (2.12)

m(n) : J n(ar)arn da, (2.13)

13

A -

WIN=and where

a 0r = particle radiusn(or) = particle distribut4on function

riim(n) = the nth moment of the particle distribution function

0.0 av< 0.50 0.0 < Rep < 2.0.

2.7 Heat (Mass) Transfer Between a Cloud of Droplets and the Surrbunding

Fluid

In our case of interest, the interfacial transfer of heat and mass

between a single droplet and the surrounding fluid is primarily due to

forced convection. Free convection and radiative (heat) transfer are

neglected. The transfer rates may be determined by the following.

Heat Transfer Rate = hh A(T - T )(2.14)

Mass Transfer Rate = hm A(Cws - C wg)where

hh = average heat transfer coefficient for the droplet (cloud)

hM = average mass transfer coefficient for the droplet (cloud)

A droplet (cloud) surface area

T = temperature

Cws = water vapor concentration at surface droplet (cloud)

Cwg water vapor concentration in fluid flowing around droplet

(cloud).

When the relative vzlocity between a single droplet and the

surrounding fluid approaches Eero the following relationship is used toidetermine the transfer rates (Ref. 29).

Nu = 2.0 3

Sh = 2.0 (2.15)

14T

where

, = Nusselt number hh ds/k

Sh = Sherwood number = hm d s/Db

k = thermal conductivity of the fluid surrounding the droplet

(cloud)

Db = binary mass diffusivity.

The convective interfacial heat and mass transfers are a function of

the respective P6clet number, and the internal circulation of the droplet.

The Peclet numbers are defined as follows:

Peh = ReD x Pr =I~a V fd/a (2.16)

Pem = Rep x Sc = IVg - Vp~ds/Db (2.17)

where

Peh = heat transfer Pecl t number

Pem = mass transfer Peclt number

Pr = Prandtl number = vi /p g

Sc = Schmidt number = vg /pg Db

= thermal diffusivity of the fluid surrounding the droplet

(cloud).

Again assuming ds = 1 mm and a relative velocity of 10 m/sec, Peh = 498,

and Pem = 391 at standard atmospheric condiLions.

When the particulate phase changes from discrete droplets to bubbles

the internal circulation varies from zero to a maximum. Fluid droplets

are considered to have an intermediate amount of internal circulation.

The presence of surfactant (surface reactant) impurities contaminates

15

the gas-fluid particle interface and also tends to reduce the internal

circulation in droplets and bubbles. This in turn considerably reduces

the rates of interfacial transfer.

References 30 and 31 provide a summary of a large number of analytic

and correlated results for s-ngle particle interfacial transfer. Three

such results follow.

Friedlander (Ref. 32).

Peh Peh2 1Nu = 21 + 4h + h

pem Pe 2 (2.18)Sh = 2[l + +.J!1. + ]

These results apply for no internal circulation, small P6clet numbers,

and small particle Reynolds numbers.

Levich (Ref. 33).

Nu = 0.998 Peh1/3 + 2

(2.19)

Sh = 0.998 Pe mf3 + 2

Levich's results apply for no internal circulation, large Peclet numbers,

and small particle Reynolds numbers.

Ruckenstein (Ref. 34).

Nu = 0.895(Z) Pehk + 2

(2.20)

Sh = 0.895() Pem + 2

where

Z 2(l + a) (2.21)

9 : /(Pp + E) (2.22)

= "interfacial viscosity" due to adsorbed surfactant impurities.

16

These results apply for toderate to strong internal circulation, large

Peclet numbers, and small particle Reynolds numbers.

When the relative velociy between a cloud of droplets and the

surrounding fluid approaches zero the following relationship may be used

to determne the transfer rates (Ref. 31).

Nu = 2/(1 - ovV 3

(2.23)Sh = 2/(I - av/3

The Mus elt and Sherwood numbers are defined in the same manner as for

the single droplet case, but note that the fluid surrounding the cloud of

particles may well include other particle clouds and therefore the

thermal conductivity, thermal diffusivity, and binary diffusion coeffi-

cient must be modified accordingly.

The interfacial convective heat (or mass) transfer between a cloud

of droplets and the surrounding fluid is not only a function of the

Peclet number (Peh,Pem) and the internal circulation of the particles.

It is also a function of particle concentration and particle size

distribution. These reflect the influence of both particle-particle

interaction and hydrodynamic fluid-particle interaction. References

(35-44) provide a short bibliography on this subject. Two examples of

correlations are given here, one due to Reference 36 and the other due to

Reference 41.

Pfeffer (Ref 36). -

Nu = 1.26 - v / Pe' 3 + 2

Sh =1.26 + 2XP- (1 - v13)

17

I.n

where

Y 2 + 28 + v"3(3 - 28) (2.25)V

X = 3 + 28 + 2a v 3(1 .(2.26)

Note that the relative velocity used to define the cloud Peclt

number is defined as the difference between the gas flow velocity and the

mean velocity assigned to the cloud of particles. Refer to the second

consideration listed in Section 2.

Pfeifer's results apply to the flow of an ensemble of uniform size

particles with no internal circulacion, a large Peclet number, and a

small particle Reynolds number.

Yaron and Gal-or (Ref. 41)

1.1284 Pe2h (2.

(1• ) 2.8852Fjzp) (1- /)Nu = [1 F(p,p)] + (227)

(l - v) Re (1 -av)(.

1. 1284 Pe h 285 2Sh = (I- a l Rp2 8 5 F(p,p)] + (I-a )(2.28)

1 5 P ) ]where F(p,p) : [I + /[I + 2

These results apply to the flow of a cloud of uniform size particles

with no restriction on the amount of internal circulation, a large Pecl6t

number, and 50 < Rep f 1500.

As previously mentioned, we are interested in the variation of the

thermal conductivity of the overall mixture with particulate volume

fraction, av* Gotoh, Reference 45, recommends without restrictions the

use of the following equation for evaluating the effective thermal

conductivity of two-phase heterogeneous flow.

18

1- k ke kg (2.29)

where

ke the effective thermal conductivity of the mixture.

Yamada and Takahashi, Reference 46, have conducted an interesting

study on the vari'ation of the effective thermal conductivity of suspen-

sions with particle shape, orientation, dimension, and surface condition.

k

19

Zl! WT

L

3. PERFORMANCE ESTIMATION

In light of the discussion in Section 2, a multi-stage axial

comoressor operating with a two-phase fluid is studied best by dividing

it into three parts as follows:

1) The low pressure stages in which no substantial change of phase

(from liquid to vapor) has occured;

2) The intermediate stages in which there is a complete change of

phase; and

3) The high pressure stages in which one has essentially an air-

vapor mixture.

In each group there is a change of performance, efficiency, and surge

margin.

In view of the uncertainties on transport properties of particulate

two-phase flow, we have concentrated in the present investigation on the

performance of a compressor operating with air-vapor mixture. This

effort is divided into the following:

(1) Performance calculation of a single stage air compressor,

PURDUE 001 & 002 Compressors

(2) Performance calculation for the multi-stage compressor, N.G.T.E.

#109 Compressor

(3) Performance calculation for the P&W TF-30 Compressor.

Each of the forementioned are evaluated using temperatures corresponding

to: (1) design conditions - 288 K, (2) air containing water vapor in the

form of dry steam - 373 K, and (3) air before the latent heat is absorbed

due to evaporation - 429.04 K or 484.79 K.

20

3.1 Single Stage Compressor

A single stage compressor was desinged with the following design

parameters:

mass flow = 20 kg/sec

stage temperature rise = 20C

r.p.m. = 9,000

mean blade speed = 180 m/sec

Both a constant -eaction (PURDUE 001) and a free-vortex (PURDUE 002)

compressors were designed.

The performance of the single stage compressor was calculated for

operation with the following working fluids:

a) air at 14.70 psia pressure and 429.04JK, 373.00 K, and 288.00 K

temperature.

b) air mixed with 5.0% steam (by weight) at 14.70 psia and 373.0°K

temperature. The results are presented it, Figs. 3.1-3.13.

3.2 Multistage Compressor: N.G.T.E. #109

The design details for this compressor can be found in Ref. 47.

The performance of the compressor was calculated for operation with

the following working fluids:

(a) air at 14.70 psia pressure and 484.79'K

(b) air mixed with 10% steam (by weight) at 14.70 psia pressure and

373.0°K temperature

The results are presented in Fig. 3.14.

21

4 3.3 P&W TF-30 Compressor

In view of the lack of consistent design and experimental data, it

has not been possible to carry out performance calculations, although

plans have been made to carry out the following calculat-,ns:

(a) H.P. compressor performance with air mixed with 10% by weight

steam.

(b) L.P. compressor performance in the initial stages with air

mixed with 10% by weight water in uniform droplet form.

(c) Stage 10 with sudden evaporation of 5% water and with air mixed

with 10% by weight steam, although this is in principle equivalent

to the calculation described under 3.1.

2II

22

4. COMPRESSOR AEROTHERMODYNAMICS

The compressor aerothermodynamics developed here is based on

References 48-56. However, the working fluid is assumed to be an air-

water mixture in which there can arise . change of phase from liquid

water to vapor. In view of the fact that the water droplets may be

subjected to gravitational forces, especially in inter-blade spacings and

in non-rotating blade passages, gravitational forces .,ave been included

in the three-dimensional two-phase flow equations.

In the absence of gravitational forces, it is proposed to develop a

compressor performance calculation method based on the axisymmetric flow

assumption.

4.1 General Considerations

a) State Relations

i) GAS PHASE: The gas phase is considered to be a mixture of air

and water vapor that behaves as a perfect gas. Therefore,

Pg = pg Rg 1 (4.1)Pg =presure ssoiate wit tg g g sphs

where

P9 = pressure associated with the gas phase

Rg = gas phase gas constant.

ii) LIQUID PHASE: The liquid phase is assumed to have a constant

density,Pp 0.988 gm/cm 3. (4.2)

I It is assumed that when a reaches a value of 0.40 for tap water and

air (0.90 for "pure" water and air) at a pressure of one atmosphere a

film arises on the casing wall (Section 2.5).

23

4

b) Stress-Strain Relations

i) Pressure: The mixture pressure (gas continuum + liquid

"continuum") is assumed to be equal to the gas phase pressure,

Pg. It is assumed that the water droplets do not contribute

to the local pressure due to the lack of randomness in local

particle motion.

ii) Viscosity: A turbulent, isotropic, effective coefficient of

viscosity based on gas-phase Reynolds number is assumed

throughout. The water droplet content is therefore not

accounted for in estimating the mixture viscosity.

iii) Viscous force on droplets: The viscous force on droplets is

based on a Reynolds number calculated with respect to the gas

phase kinematic viscosity (effective), the relative gas

velocity, and the local statistical mean diameter of the droplet

(Section 2.6).

iv) Surface tension: The droplet breakup is based upon a critical

value of Weber number (of the order of ten) and the surface

tension for water is taken to be

a = 67.91 dynes/cm. (".3)

c) Water Droplet Fraction

The particulate droplet mass fraction is defined as follows:

a ((m:/VOW/(m/Vol m = mp (4.4)

where

a m : particulate droplet mass fraction

m mass

Vol Volume

( )- : pertaining to the mixture as a whole.

24

The particulate droplet volume fraction is defined as follows:

av Volp/VOl. (4.5)

E-.panding Equation (4.4) and substituting (4.5) an expression is

obtained relacing o;v and am

am :v pp (av[Pp P g + Pg) (4.6)

or

v M am pg/(pp -amLP p - pg]. (4.7)

0v is a function of the statistical mean diameter and number

density of the particulate cloud.

IV ds3 Ns (4.8)

where again

ds = the Sauter statistical mean diameter of a cloud of5Idroplets

Ns = the statistical number density, i.e. the statistical

number of ds size particles per unit volume of overall

mixLure.

4.2 Conservation Equations

a) Assumptions

i) The gas phase is assumed to be a perfect gas continuum.

ii) The particulate liquid phase is assumed to be a continuum that

does not contribute to the local pressure. Two exceptions to

3this continuum assumption are the determination of a) the

viscous interaction force between the particulate liquid and

gas phases, and (b) the interfacial heat and mass transfers.

25

iii) Mass conservation is assumed for each of the two phases, the

gas phase and the particulate liquid phase, allowing for

phase change.

iv) The liquid mass that is vapori;:ed is assumcid to be instantly

and intimately mixed with the gas phase, i.e. large scale

vapor concentration gradients and the momentum of the

vaporized mass are neglected.

v) On the scale of droplet dimensions, there are local variations

in velocity, temperature, and vapor concentration. The evapora-

tion of individual droplets is calculated as a process

dependent upon such local properties.

vi) When axial symmetry of the flow is assumed, it is necessary to

treat the gravitational body force as being negligible.

b) Description of Coordinate Systems

The equations have been deduced using two coordinate systems,

intrinsic and cylindrical.

i) Intrinsic Coordinate System (Figure 4.1)

The m,e,n intrinsic system is a local, right hand, orthogonal

system. The m-direction lies tangent to the local streamline

direction, and the e-direction is orthogonal to the m-r plane

where r is the radial vector defined in Figure 4.1. The n-

direction is normal to the m-direction in the local meridional

plane.

ii) Cylindrical Coordinate System (Figure 4.2)

The cylindrical system is fixed to the axis of rotation of the

compressor. The z-direction lies along the axis of rotation

26

I

in the direction of the bulk flow and r is in the radial

direction. The e-component is in the circumferential direc-

tion defined positive with respect to a right-hand r,e,z

orthogonal coordinate system.

iii) Rotation of the Machine

It is more convenient to work in coordinate systems which are

stationary with respect to a rotating blade row. Hence fluid

velocities for a rotating machine are defined as follows:

(1) Intrinsic Coordinates

SW + U (4.9)

where

= absolute velocity vector

= relative velocity vector

= local blade rotational velocity vector = U m 0 +n

UmUo,Un = the m,e,n components of U, respectively,

but

Um = f04J2)

U6 = fe(U, 41,P2) (4.10)

Un = fn(U,41,"2)

where

U = the magnitude of the local blade rotational velocity

vector

UmU0,Un : the magnitudes of the m,e,n components of U

! = the angle between the radius vector and the m-direction

2 = the angle between the radius vector and the n-direction.

27

Therefore,V = W + U Vp = 1IT + Ugmn gIn Inp p m

Vgo 1go + U V p0 = Wpo + [ (4.11)

Vgn Wgn U n V = W + U n OJ

(2) Cylindrical Coordinates

Again -+-

where similarly

U Ur + U 0 + U Z. (4.12)

But

U r = = 0. (4.13)

Therefore

U =U 0 (4.14)

and U Ue =rw (4.15)

where

w = angular velocity of the rotating blade row.

As a result,

Vgr Wgr Vpr Wpr

V go Wgo + rw VP6 =Wo + rw (4.16)

Vgz Wgz Vpz Wpz

c) Flow Equations

The following flow equations are deduced in both sets of

coordinate systems for three-dimensional as well as axisymmetric

flow. They consist of the following:

28

i) Mass Conservation equations

ii) Momentum Conservation equations

iii) Energy Conservation equations

iv) Variation of ov with respect to the three coordinate

directions

v) Radial Equilibrium equations

The equations are presented in Appendix II as follows.

II.1. Three-dimensional Flow Equations in Intrinsic Coordinates

11.2. Axisymmetric Flow Equations in Intrinsic Coordinates

11.3. Three-dimensional Flow Equations in Cylindrical Coordinates

11.4. Axisymmetric Flow Equations in Cylindrical Coordinates

29I

29

U..-. i

5. DISCUSSION

i- f The study presented in this Report is concerned with the operation

of an aircraft compressor with an air-water mixture ingested into the

engine either from puddles on runways or from rain at altitude.

The water in the air-water mixture may be found in several states

(discrete droplets, film of water, evaporating droplets and steam)

between the inlet and outlet of a large pressure ratio multi-stage

machine, such as the TF-30 compressor. Agglomeration and break-up of

droplets, the transport processes (mass and heat) between the droplets

and the surrounding air and the drag of droplets are not phenomena that

are well understood quantitatively. They are central to gaining an

understanding of the changes in the state of water in the system.

The presence of water in air also affects the compressibility of

the mixture and therefore the local Mach number at any location in the

compressor becomes a function of the state and content of water in the

mixture.

The rotating flow field of a compressor has a strong influence on

the state and distribution of water since centrifugal forces and Coriolis

forces affect both discrete droplets and films of water.

Discrete droplets are also subject to gravitational influence, at

least at low rotating speeds and in inter-row spacings and stators.

In examining the performance of a multi-stage compressor with an

air-water mixture flow, it seems most useful to divide the compressor

into groups of stages on the basis of the state in which water isexpezcted to be found in each group. In doing this, there arise at least

two important uncertainties: (1) the formation of a film at the casing

wall and (2) the gradual evaporation of the droplets.tN

30

- R

AI Some preliminary investigations conducted on the performance of a

single stage machine with air-water droplet mixture flow have revealed

that the water droplets cause both a reduction in mass flow as well as a

slight change in efficiency, mainly due to incidence and Mach number

effects.

The performance of stages where the droplets are expected to be

evaporating cannot be estimated with any degree of accuracy based on

overall stage characteristics. Nevertheless, it is possible to argue

that the stages where the largest amount of evaporation occurs must show

the largest change in performance on account of the sudden fall in air

temperature due to latent heat absorption by the water.

When there is an air-steam mixture flow, it is possible to make a

rough estimate of the change in overall stage characteristics. It is

found, both in a single stage compressor and in a multi-stage machine,

that there is a noticeable deterioration in performance both in efficiency

as well as in the surgeline.

In order to perform more satisfactory calculations, it is imperative

that one proceeds to set up the equations of aerothermodynamics for air-

water mixture flow in a compressor. This task has been completed both in

general three-dimensional coordinates as well as in axisymmetric

coordinates. Such equations involve several quantities that can only be

estimated very roughly and hence can perhaps be treated only in a para-

metric form: (1) volume fraction of the water droplets, (2) statisticalmean di ameter and nlumber 'ens Iity IU

an...... of droplets, (3) mass, momentum and

go energy transfer between the droplets and the air, and (4) degree of mixing

between evaporated water and the surrounding air.

31

The equations for air-water mixture, when compared with the standard

compressor flow equations for air flow, reveal several ways 'in which

computational programs set up for air flow can be adapted for use with

air-water mixture flow.

In examining the problem from the point of view of alleviating the

effects of water ingestion, one can arrive at several methods of attack

on the problem:

(a) Removal of as much water as possible at the casing walls.

(b) Bleed-in of air into the latter stages of the compressor from

another stream.

(c) Variable geometry engine with adjustable air flow paths.

5.1. Recommendations

The recommendations may be divided broadly into two categories:

(i) Further diagnostic studies.

(ii) Future studies.

5.1.1. Further diagnostic studies

(1) Estimation of performance of selected multi-stage machines

(incorporating bleed-in and bleed-out) on the basis of stage character-

istics when there is a simultaneous change in temperature from the design

value and an addition of steam.

(2) Estimation of the effect of bypass ratio on the performance of

compressors with two-phase flow.

(3) Study of two-phase flow in a diffusing passage.

(4) Adaptation of simple, existing computational programs for air

flow to air-water mixture flow on the basis of parameterized droplet

dynamics and transport processes for the study of a selected multi-stage

machine.32

(5) A flow visualization study on a multi-stage compressor to

establish the water droplet dynamics when water is ingested into the air

stream at entry to the compressor

(6) A flow visualization study on a multi-stage compressor with

selected inlet configurations to establish the water droplet dynamics

when water is injected into the air stream at entry to the inlet.

(7) A flow visualization study on a rotating tire in a wind tunnel

equipped with a moving belt floor to establish the influence of a few

selected tire parameters on the jet created by the motion of the tire.

5.1.2. Future studies

(1) Development of a comprehensive computational program for air-water

mixture flow through an axial compressor with appropriate droplet dynamics

and transport coefficients.

*(2) Analysis of the performance of compressors using the computational

program developed to establish the scope and implications of variable

cycle/geometry schemes.

(3)(a) Study of bleed-out of water in early stages.

(b) Study of bleed-in of air into the latter stages.

(4) Determination of classes of compressors to which different

classes of controls can be applied.

(5) Study of the effects of engine-airframe integration parameters

on the utilization of different kinds of controls.

(6) Study of the water jet created by a moving aircraft tire on a

wet runway in relation to the location of the engine inlet.

(7) Determination of the effect of water vapor on after-burner

stability, particularly in relation to the fuel vaporization and combus-

tion characteristics.

33

6. LIST OF REFERENCES

1. Willenborg, J. A., Jordan, J. E., Schumacher, P. W. J., Hughes, P. L.,

Willocks, H. J., Crabtree, C. F., Shafer, R., Connell, R. E., Dean, R.,

Scott, D. L., and Stibich, M. A., "F-Ill Engine Water Inqestion

Review," F-Ill System Program Office, Wright-Patterson Air Force Base,

Dayton, Ohio, October 31-November 10, 1972.

2. Holmboe, J., Forsythe, G. E., and Gustin, W., Dynamic Meterology,

pp. 57-60, John Wiley and Sons, Inc., New York, 1945.

3. Green, H.L. and Lane, W. R., Particulate Clouds, Spon Ltd., London,

1964.

4. Fuchs, N. A., The Mechanics of Aerosols, Pergamon Press, Ltd.,

London, 1964.

5. Soo, S. L., Fluid Dynamics of Multiphase Systems, Blaisdell Publishing

Company, Waltham, Massachusetts, 1967.

6. Wallis, G. B., One-dimensional Two-phate Flow, McGraw-Hill, Inc.,

U.S.A., 1969.

7. Rudinger, G., "Fundamentals and Applications of Gas-Particle Flow,"

Paper prepared for the meeting of the AGARD Fluid Dynamics Panel in

Rome, Italy, September, 1974.

8. Briggs, J., "Probabilities of Aircraft Encounters with Heavy Rain,"

Meterological Magazine, Vol. 101, pp. 8-13, 1972.

9. Methven, T. J. and Fairheed, B., "A Correlation between Rain Erosion

of Perspex Specimens in Flight and on a Ground Rig," A.R.C. Technical

Report 21,090, C.P. No. 496, 1960.

10. Hays, D. and Browne, A. L., Ed., "Symposium on the Physics of Tire

Traction," General Motors Corporation Research Laboratories, Plenum

Press, New York, 1974.

34

11. Collier, J. G. and Wallis, G. B., Two-phase Flow and Heat Transfer,

Vol. II, pg. 405, Dept. of Mechanical Engineering, Stanford

University, Stanford, California, 1967.

12. Wallis, G. B., One-dimensional Two-phase Flow, pp. 378-380, McGraw-

Hill, Inc., U.S.A., 1969.

13. Ibid, pp. 376-377.

14. Ibid, pp. 265-267.

15. Swain, C. E., "The Effect of Particle/Shock Layer Interaction on Re-

entry Vehicle Performance," AIAA Paper 75-734, June, 1975.

16. Ergun, S., "Fluid Flow through Packed Columns," Chemical Engineering

Progress, Vol. 48, No. 2, pp. 89-94, February, 1952.

17. Ingebo, R. D., "Drag Coefficients for D-oplets and Solid Spheres in

Clouds Accelerating in Airstreams," NACA Technical Note 3762, May,

1956.

18. Marble, F. E., "Mechanism of Particle Zollision in the One-

dimensional Dynamics of Gas-Particle Mixtures," The Physics of

Fluids, Vol. 7, No. 8, pp. 1270-1282, August, 1564.

19. Fuchs, N. A., The Me':hanics of Aerosols, pp. 21-159, Pergamon Press,

Ltd., London, 1964.

20. Soo, S. L., "Dynamics of Multiphase Flow Systems," Industrial and

Engineering Chemistry Fundamentals, Vol. 4, No. 4, pp. 426-433,

November, 1965.

21. Soo, S. L., Fluid Dynamics of Multiphase Systems, pp. 185-244,

Blaisdell Publishing Company, Waltham, Massachusetts, 1967.

22. Tam, C. K. W., "The Drag on a Cloud of Spherical Particles in Low

Reynolds Number Flow," Journal of Fluid Mechanics, Vol. 38, Part 3,

pp. 537-546, 1969.

35

.JmLl P .T

23. Rudinger, G., "Effective Drag Coefficient for Gas-Particle Flow in

Shock Tubes," Journal of Basic Engineering, Transactions of the ASME,

Series D, Vol. 92, pp. 165-172, March, 1970.

24. Pai, S. I. and Hsieh, T., "Interaction Terms in Gas-Solid Two-phase

Flows," Zeitschrift Fur Flugwissen Schaften, Vol. 21, Heft 12,

pp. 442-445, 1973.


pp. 12-14, Paper prepared for the meeting of the AGARD Fluid

Dynamics Panel in Rome, Italy, September, 1974.

26. Hamed, A. and Tabakoff, W., "Solid Particle Demixing in a Suspension

Flow of Viscous Gas," Journal of Fluids Enineering, Transactions of

the ASME, Series I, Vol. 97, No. 1, pg. 106-il1 March, 1975.

27. Law, C. K., "A Theory for Monodisperse Spray Vaporization in Adiabatic

and Isothermal Systems," International Journal of Heat and Mass

Transfer, Vol. 18, pp. 1285-1292, 1975.

28. Batchelor, G. K., "Brownian Diffision of Particles with Hydrodynamic

Interaction," Journal of Fluid Mechanics, Vol. 74, No. 1, pp. 1-29,

1976.


pp. 14-15, Paper prepared for the meeting of the AGARD Fluid

Dynamics Panel in Rome, Italy, September, 1974.

30. Griffith, R. M., "Mass Transfer from Drops and Bubbles," Chemical

Engineering Seience, Vol. 12, pp. 198-213, 1960.

31. Yaron, I. and Gal-or, B., "Convective M~iss or Heat Transfer from

Size-distributed Drops, Bubbles, or Solid Particles," International

Journal of Heat and Mass Transfer, Vol. 14, pp. 727-737, 1971.

36

32. Friedlander, S. K., "Mass and Heat Transfer to Single Spheres and

Cylinders at Low Reynolds Numbers," American Institute of Chemical

Engineers Journal, Vol. 3, pg. 43, 1957.

33. Levich, V., Physiochemical Hydrodynamics, Prentice-Hall, Englewood

Cliffs, New Jersey, 1962.

34. Ruckenstein, E., "On Mass Traribfer in the Continuous Phase from

Spherical Bubbles or Drops," Chemical Engineering Science, Voi. 19,

pg. 131, 1964.

35. Hsu, N. T., Sato, K., and Sage, B. H., "Material Transfer in

Turbulent Gas Streams," Industrial and Engineering Chemistry,

Vol. 46, No. 5, pp. 870-876, May 1954.

36. Pfeffer, R., "Heat and Mass Transport in Multiparticle Systems,"

Industrial and Engineering Chemistry Fundamentals, Vol. 3, No. 4,

November, 1964.

37. Kunii, Daizo and Suzuki, Motoyuki, "Particle-to-Fluid Heat and Mass

Transfer in Packed Beds of Fine Particles," International and

Journal of Heat and Mass Transfer, Vol. 10, pp. 845-952, 1967.

38. Gal-or, B. and Walatka, V., A Theoretical Analysis of Some

Interrelationships and Mechanisms of Heat and Mass Transfer in

Dispersions," American Institute of Chemical Engineers Journal,

Vol. 13, No. 4, pp. 650-657, July, 1967.

39. Gal-or, B., "Coupled Heat and Multicomponent Mass Transfer in

Particulate Systems with Residence Tiwe and Size Distributions,"

International Journal of Heat and Mass Transfer, Vol. li, pp. 551-

565, 1968.

37

VI

40. Waslo, S. and Gal-or, B., "Boundary Layer Theory for Mass and Heat

Transfer in Clouds of Moving Drops, Bubbles, or So. i Particles,"

Chemical Engineering Science, Vol. 26, pp. 829-838, 1971.

41. Yaron, I. and Gal-or, B., "High Reynolds Number Fluid Dynamics and

Heat and Mass Transfer in Real Concentrated Particulate Two-phase

Systems," International Journal of Heat and Mass Transfe.r, Vol. 16,

pp. 887-895, 1973.

42. Hughmark, G. A., "Heat Transfer in a Fluidized Bed of Small Particles,"

American Institute of Chemical Engineers Journal, Vol. 19, No. 3,

pp. 658-659, May, 1973.

43. Yaron, I. and Gal-or, B., "Low Peclet Number Mass or Heat Transfer

in Two-phase Particulate Systems," American Institute of Chemical

Engineers, Vol. 19, No. 3, pp. 662-664, May, 1973.

44. Labowsky, M., "The Effects of Nearest Neighbor Interaction on the

Evaporation Rate of Cloud Particles," Engineering and Applied

Science Dept., Yale University, New Haven, Connecticut, August, 1975.

45. Gotoh, K., "Thermal Conductivity of Two-phase Heterogeneous

Substances," International Journal of Heat and Mass Transfer,

Vol. 14, pp. 645-646, 1971.

46. Yamada, E. and Takahashi, K., "Effective Thermal Conductivity of

Suspensions - 1st Report: Analysis of the Effectc of Various

Factors by the Electrolytic-Bath Method," Heat Transfer - Japanese

Research, Cosponsored by The Society of Chemical F;1,ineers of Japan

and The Heat Transfer Division of ASME, Vol. 4, No. 1, January-I

March, 1975.

38

iF

* 47. Carter, A. D. S., Turner, R. C., Sparkes, D. W., and Burrows, R. A.,

"The Design and Testing of an Axial-Flow Compressor having Different

Blade Profiles in Each Stage," Communicated by the Director-General

of Scientific Research (Air), Ministry of Supply, Reports and

Memoranda No. 3183, November, 1957.

48. Shapiro, A. H., The Dynamics and Thermodynamics of Compressible

Fluid Flow, Vol. I and II, The Ronald Press Company, New York, 1953.

49. Schlichting, H., Boundary Layer Theory, Pergamon Press, Ltd., New

York, 1955.

50. Oswatitsch, Klaus, Gas Dynamics, Academic Press, Inc., New York, 1956.

51. Emmons, H. W., Fundamentals of Gas Dynamics, Princeton University

Press, New Jersey, 1958.

52. Daily, J. W. and Harleman, b R. F., Fluid Dynamics, Addison-Wesley

Publishing Company, Inc., Reading, Massachusetts, 1966.

53. Wooten, D. C., "The Attenuation and Dispersion of Sound in a

Condensing Medium," Ph.D. Thesis, California Institute of Technology,

Pasadena, California, 1967.

54. Marble, F. E., "Dynamics of Dusty Gases," Annual Review of Fluid

Dynamics, pp. 397-445, Annual Reviews, Inc., 1970.

55. Sonntag, R. E. and Van Wylen, G. J., Introduction to Thermodynamics:

Classical and Statistical, John Wiley and Sons, Inc., New York, 1971.

56. Wennerstrom, A. J., "On the Treatment of Body Forces in the Radial

Equilibrium Equation of Turbomachinery," ARL 75-0052, April, 1974.A 57. Lighthill, M. J., "Mathematics and Aeronautics," Journal of the Royal

Aeronautical Society, Vol. 64, No. 595, pp. 375-394, July, 1960.

58. Taylor, G. I., "Generation of Ripples by Wind Blowing over a Viscous

Fluid," The Scientific Papers of G. I. Taylor, ed., G. K. Batchelor,

Vol. 3, pp. 244-254, Cambridge University Press, 1963.

39

FIGURE S

40

_____FIGURE 2.1

4LL fIp:rLt T41- ii It- _T_ TS j1- 1 TL..

+ 1-41F+

44 -4-4 4 Il z- -_ _

_ 4 _ I

HL

:P41

4J J-i4~~~-1 #1 - '

Lz

Lz~

42

I 'f il**

t 11

I

:4 /A1 -i4L IH'

A- _4-4--- -j I A

TL -- I- AL T'~IML I.

JJ-j F-1 J i

I~lI H I _ __ -

I tI T

43

hiU

Ll IT

_ __ __-.--JA---_ __ 4--

I L L Jbz 11AILL ht-'--F41VLLt CVL ___J _

I1~- _ _LM -~~.LL -..- - **11

I24~~ 1117

e.% I I Y_ LI4I

9A'F i 7 Az 9 Al.~. Izt~zzi±I'z'l~l~ ~,i~Kj:fl

:4:f~z~zI~ip~z~z~ ~ jIL __

14

__ II

4 -

-A-

I L

CC

C3 j IJJ4 1)

'~~ I J I i )-

-3 4J

01. In

-4j

iii~C -- 4

___ A

n. N aMA ~ ~ - a

____ ~ ~ ~ T- ___ __ L_____ J _______I---.

4~f45

41 ~ CA _

7I4 4iMit IC_ -_

1j1.p9 ~k dP1i ~4ii_ ___

CD V1_ * _ In~.

j~l I __ ~ ION,

I _ HI-I __ _ i.~~t~]zi.lyl ~IpfL IIJIL- 1~ .H ±

__ _ - i- -- I1JI[ f I.rjI 1III+ 1 1 4L __ezrz

-I _ -I- ~[ .jiii c ,

06 *-i_-

I - -46

I

--I- - 1

T~~~~~ f :-i -i, t l_

cr1- il I

R _ I "F-F I -

" II1 _

11 !I I

i_!_I T[ -ILI

- - . .. U

Elll_ f - I , - .....

ii a I ~ __4Z

~ t IL~i ~IiIJ!L 7 1 lap T

ill { j ~-J-f-V N l :~ 'I 1_47 -~ ~ (

I- I 7t

I if

4 1

4~ Ll

~~ UiI

:~-.-T 1- -i* I I

S(DI L 4-1 II I

II -4

d' I f

rI Ln1FT IIJJ7I-

L L

-- I

AL

v i2 f ilYiL2i!Ij~~j ~f~ Ij I ~ I' I

~t VCIis-f L - j

+ 41!~f~;0I0 I

IL 1 IL I I I I

to I--

-J- -L Il1iii I i I IIii

L1117 LIJLL L iIi Ii

I iI lo d T i '0 Hf T~1t I-, .~~ ~ ~-F-- 9 II 49

i4-1

LIp E, f~Y

I-

- ~ 4 if -A - I i ~ i

74-

I~4 I' II I

INK i I I I IN

I Ii'I CDt.

_JJL

I I I__

I~~ - *TI

~ ~ ~ r~i-51

JM

_ I , II t

So- 1 -4 ---

I W 0C2L-n

J

.Lr.. 1L__ _

AW mV.IF - -

--f7-1+-_ - -

zzf:~ -'~ Jj 1+ - - --- _c IVf irs

-- 1 t I 7-ll I

t It . 4

tttt F-

1I1 I I I <

;; I f

53

-44-

U~)-7

- -t

C 1

-4-H- ii L~) L .

- IL LL~t

II II

c1ii0IIY"L 15i7

- -r-H-Lfl+ I -o dT

-LKLL -TrLT- 1 Tj~f. i44jj; ~54

-TA

tt j____ 1- Qj-~

J- 1-4~ _ _ _ __ _

4H

ItI

u ,; F--

-- i--4- --- ~ 1 ____31

!-- IF _

Ll ____ c

--

I-t

55 -

hiI

h -4

-1 4

Ti ai1'I

_jt1 -4-.

as.-W_4 1v 114t

n 47I--L.--.--.

-

ii- I RUL~-pTM.~

564

- - i -FFJ- -z--_ - - L

461-___

TK -4K .r t 4AAIJ~4'I _ _I i-.t

2di L.L4L H I t-Pu t 4* 1I__- - - - - i

L- - -T L - - I-___ j57

i~ ii~ j~tjivjE~fJ1j]

-F - -- I- VLL

El ';I I I?

I-] I fIji 17 coi-Q.4 +TLH

~jj---~---~ ~Ij7~ __2Lif:~ D

_C I _

~jII

----IU!2~ ___

0 58

- _ _-

I Im en IY

Ii Li Intn~;L ~~ 44- L

III

- _ -. +II ~~sICU

-! -~ i~' 7:jfTJziirI - I

CD -. C v)

-t~c 11119i

FIGURE 3.9 _

P U R Oa9 $ J -t i

'14 ia

y I I j I j

:: -. -4vrI 11111-11- _ -D-EL

ea__~~ I hf~ lis~ s I

1 " 1 1 1 1 1T1 +

pl. p1 1' 1 -w.-i

FIGURE 3.10a

S ....... | Ai _q e v.;. Ra i, Rat,

f o 1 blRi Ac ----- --s- I-- ++--

.[..... . hfIE . . . . - - i . -- - _

,., _ _ _ _- _,-,-+4Ih -I.. . .. ..I .. . '

]I

7 1 1 1 l 1 1 11 l 1 2 .

-ri---- !,-_ _ ------- 4=~. .. ... H - -- - - --

._u, r! ____ _

-

--

1 6

_ :_:i i :1441... I } +'°-, .,l'i61

FIGURE 3.10b

ljPU DEO 0; OMRISSO RC0S-AN~T EJCTOp_ _i .-_, E ___a -T ja- _ -_

t I __1

I~~~~ 7_1------~~I

k-i I Y i II i iI,=1 I ~1 V _

V if~~izrfi -' _

71 1 _1 I I

MD M - iu I-__j .L ~ 1 - 62

_ I I URI 3. 1 a

-I---'-

ittt-1-1~ Th-

-L 4

1 1 -

H~~~t -5,LF.

-T I I _i 2j1T-J7

I VI

--

I~~~L -11KL '63

FIGURE 3.11b

_ _Q _.. a LLL. 3K

-------- * --- 1.

zjzzi I_ _ _ lejj.{i~cL L~KIzTzji17 Iitil I f-----

IV-t-LN _!Ab --.

LT-i-i*- T_ 1 ±111 1 -

I~~~~~' -171 1i ~I~tlZV! ill

_ .71 1 -' 1 1 'F1I'IR DI -r t 1z

liii I~ 64

A-P

FIGURE 3.12PURDUE 001 COMPRESSOR -CONSTANT REACTION and PURDUE 002 COMPRESSOR -FREE VORTEX

TI0 hti2IT7 I

-0- ! I luYlv.41 vi- s. Lpad [li

I I jT + -TII-J

Ii v qiie vue

II t

t6, 1- Tj

j~mi 1 Ft Ad hl t4

I '~' ijI~I iji Iii

Now I

PURDUE 001 P~~L FIGURE 3.13 UJO CQTE

)o gh rl V. c ci_ - B Iad lli 9 ihfk

-~ 10- - -

-~~~ --- * ----

0jII A L B AD H' T3I-I _ 7IIYiil66

FIGURE_3.14a

4.~rd R ~EEIE #109 C mpEiSOR

I- -P- c-----i-----iz

-. - - - - I r. 1 PAM

-- Ca r 's Dz tall±~ ~ I

RR- P- 01 1

4.~~~ Tt T

~1-14

UMWNA -- I U T -o -----

67

FIGURE 3.14b

FTFT- I _ I ?j.G. EEEN #1 P ME SO

. .e r .r at o v . . M a s FlowIT

1 r 4 a 4W79 K4

I V

=,

....% 1_ _ _ .4 _ _ _ _ _

-_P

-~

L..~----.-----

IE

AD--- _s _ _ --- i

NO ItI1SONL 4A!T-

16 8

!P17-

-~----------~I~ it!

_ A-1

EL--- iIzz4

- in ~ U• il:c!eI~*--I- 1168

_ FIGURE 3. 14.c

tW Ed I~ cien y v.s_ Mass Fl _ _

_1 '' 11JVKu1L~

-- ~.;zz---- ]LI

-"-1ITI 1 - --.--

~~u: 11 7 ifI L~rTi1-

LiI~t rE~~ ~ ~~7~~'TTF~Lr~ - lz69

FIGURE_3._14d

N.G." E . E ERNE # 9Cm S O- 801 - -

-- J - - -'L

JfI2~~F_ - -- ~ rp..~~wr - -- - ---- - - - i L

(-- J 1 i A S - ,- - - -- -~ - -------- I

Fo _'K '- NA I A, T -K -/H1 1

u- - A -

~ N0-r~ENINA- - - -i1Iji1{ -~--

I~-! - - --l-4--

NOI- EiU A o ij. V7 f

7111

J" rnn

0

Streamline

Three-Dimensional IntrinsicCoordinate System

L

n r

~Yc m

'-Streamline

Computing Station

Axis Of Rotation

Axisynmetric IntrinsicCoordinate System

Wennerstrom(Ref. 56)

FIGURE 4.1

INTRINSIC COORDINATE SYSTEM

71

wi

r

Axis Of Rotation

FIGURE 4.2

CYLINDRICAL COORDINATE SYSTEM

72

APPENDIX I

Estimate of Water Lifted by Wheels

An estimate of the amount of water thrown up by an aircraft wheel

from a puddle of depth equal to h can be obtained as follows.

Let the direction of motion of the wheel be x and the transverse

direction on the ground be y. Refer to Figure A.I.I.

It has been pointed out (Ref. 57) that the relative flow between

the wheel and the water is hypersonic in the sense that the Froude number,

Uc/(g h) , is of the order of 100 when h is of the order of a cm. in

depth, and where Uc is the speed of the undercarriage of the wing and

hence of the order of tens of meters per second. Accordingly, one can

postulate (on the basis of blast wave analogy) a bow wave or hydraulic

jump, as shown in Fig. A.I.I, at a distance yB from the wheel.

Now, a jet of spray can be assumed from the wheel (Ref. 58) on

account of the action of the relative wind on the surface of the moving

water. The time rate of "pumping" of the water can be written as

S C (Pw (Pa ) YB (A.I.l)

where Pa is the density of air, Pw the density of water, y' the time

derivative of YB' and c, a constant equal to approximately one.

In order to obtain YB' consider the momentum of the mass, Pw YB h,

contained in a strip of water of unit width and the frictional resistance

offered by the ground, f B YB where f is the friction coefficient,

-2of the order of 10- . The two forces must balance each other together

with the momentum loss in the spray. It can then be shodn that

B- nI ) (A.I.2)

73

S;- # . . .

wherenTI 0. 5 c (pa/Pw ) / (A.1. 3)

and

n ni + f (A.I.4)2 = n

Substituting Eq. (A.I.I) into Ec. (A.I.2), the amount of water

sprayed behind the wheel per unit distance of wheel travel can be written

as follows.

mw = 4Pw(P w/P a) h2/cI. (A.I.5)

Thus, the mass of water thrown up from a puddle 0.5 meters long and 2 cm.

deep is of the order of 10 kg. during a conventional take-off. This does*

not yet account for the water sprayed forward of the wheel in its jet

stream. During a rain storm, over a runway length of 1,000 meters with a

1 cm. depth of water on the runway, the amount of water lifted off the

surface can be as much as several metric tons. Motion pictures of wheel

sprays have shown rather large pumping characteristics.

An estimate of the spray envelope from the wheel, y, and zB (the

height reached by the spray), is rather more difficult to estimate. Some

experimental studies on a typical wheel in a moving-belt equipped wind

tunnel is indicated before proceeding too far in the analysis.

74

SPRAY ENVELOPE CROSS-SECTION

MODEL OF JUMP B

y /B

K /B

y

FIGURE A.I.1

DYNAMICS OF WHEEL SPRAY

75

APPENDIX II

Flow Equations

All synbols used are defined under Nomenclature.

II.I. Three-dimensional Flow Equations in Intrinsic Coordinates

a) Mass Conservation Equations:

Gas Phase

I L_-(r [.1-"v g-J r3 We. = i- je P, \V\1 CIO)

Liquid Phase

I ~ T [& (r pp"4r 1., t £. p V~~)' rr _

-J r, (A.II.2)

+._ ,--p, U , )V- - .,

76

) omentum >"ixervation Equations:

Gas Phase

Lvf (AJ;+ Xi ___

(W% ~ )~r -c VV ~

(A.II.3)

o-component

77

____ ____ ___ __ _ __ _ a _ _ __ _ ___t_ _

FI

n-component

\ 0 c) e(A.!I. 5)

P-f~ F,3 -F-F +r,

Liquid Phase

m-component

Orn \ A.11.6)

78

I _

I|I

e-component

(A.Ii.7)

n-component

(A.II.8)

I ,*% V P,~

c) Energy Conservation Equations:

Gas Phase

Sr Io

!+ -79

Liquid Phase

- ' ( '.\~pj~ -S,

[ (Vp .. U).. j' . _: .(A. 11. 10)

d) Variation of ( with respect to the three coordinate directions:

m-direction

The equation is obtained as follows. The 0-component gas phase

momentum conservation equation is differentiated with respect to 0. Next

the m-component gas phase momentum conservation equation is solved in

terms of ( J- - o-v ). These two equations are then combined with the gas

phase mass conse'vation equation to obtain the final result.

0- __ __ Fa 5 FYI ~± 0 Fl ~i

-1

-1.

i 80

wrmI

( rC U,,, )----- r)- r C)a

(A.II.llI)

o-direction

The m-component gas phase momentum conservation equation is differ-

entiated with respect to m. Next the o-component gas phase momentum

conservation equation is solved in terms of ( I - Cr. ). These two

equations are then combined with the gas phase mass conservation equation

to obtain the final result.

-1rI, ( W, Ur- S4

81

-- _____t -3+L V, Tf J _-F ____5

r jo

A \J, + (p,

n-di rection

The equation is obtained by first differentiating the n-component

gas phase momentum conservation equation with respect to n. Then the

0-component gas phase momentum conservation equation is solved in terms

of ( I - crv ) and combined with the differentiated n-component gas

phase momentum conservation equation to obtain the final result.

- (4t . -,o +F - , \i ~r V,-.,.

9iV, + r Wr V,

82

- , I cI , 4 A1, ,1 U ., A

EW~n Stll )-(

W'3 r \ Uhb~ iA

r

(A. II. 13)

e) Equilibrium Equations:

Gas Phase

An enthalpy-entropy relation valid for the gas-phase flow can be

written as follows.

ip__ _- j I + L - T5 j s3 -Tv Ci S:' + j 0 1 (A.11. 14)

where

enthalpy of the gas phase per unit mass of the gas phase

(joules/Kgg)

c S3 = rise in entropy of the gas phase (per unit volume of mixture)

due to the action of body and surface forces on the gas phase,

excluding those forces imposed by the liquid phase (joules/*K

Vol )

83

'If = mass averaged gas phase-liquid phase temperature (OK)

cis1 = rise in entropy of the gas phase (per unit volume of mixture)

due to the action of forces on the gas phase imposed by the

liquid phase (joules/0K Vol)

c% = the heat loss from the gas phase (per unit volume of mixture)

to the surroundings, excluding the heat lost directly to the

liquid phase (joules/Vol-)

= heat loss from the gas phase (per unit volume of mixture)

directly to the liquid phase (joules/Vol-).

By definition

F1 - t (A.H1.15)

where

HCS = total enthalpy of the gas phase per unit mass of the gas phase

(joules/Kgg).

Combining (A.II.14) and (A.II.15),

(A. 11 .16)[-T°l -~r s~ + + 1 I

Combining (A.II.16) with each of the component gas phase momentum

equations (eliminating the pressure gradient term) yields the following

equilibrium results.

m-component

r

84

-~ sS1 (A. I 1.17)

e-component

SU.1, %+ F ~r~*.f{-re j~ (A.H18)

-jj

fl-component

(i~:S 4§!

85

, (A. 1.

l0

Liquid Phase

An internal energy-entropy relation valid for the liquid phase flow

can be written as follows.

U J- _ 1-p 1 -p ci e (A.II.20)

where

U internal energy of the liquid phase per unit mass of the liquid

phase (joules/Kgp)

csp = rise in entropy of the liquid phase (per unit volume of mixture)

due to the action of body and surface forces on the liquid phase,

excluding those forces imposed by the gas phase (joules/*K

Vol -)

I (P = heat loss from the liquid phase (per unit volume of mixture)

to the surroundings, excluding the heat lost directly to the

gas phase (joules/Vol).

86

By definition

Up VP.

where

P = total internal energy of the liquid phase per unit mass of the

liquid phase (joules/Kgp).

Combining (A.II.20) with (A.II.21) and expanding

d 7= Vp,, , Vp, V o r V po+ -OV'I (A.II.22)

Tp d 5 -T P1 6:

Combining (A.II.22) with each of the component liquid phase momentum

conservation equations yields the following equilibrium results.

m-component

(A. II.23)

c) , J ,do

e-component

(Vpn.4L), V'pi WP __-i r U,

(A.n1.24)

II I87

n-component

vo (S(r wp 0) S~r W9 £ pn(A.II.25)

II,2. Axisymmetric Flow Equations in Intrinsic Coordinates


Gas Phase

Liquid Phase ~L-

88

1i

b) Momentum Conservation Equations:Gas Phase

rn-component

P; W S k"Il -~ t 'w1e-r ) 2 r(A. 11.28)

O-component

(A.H1.29)

n-component

p~ -i 1 , -~(A.HI. 30)

Liquid Phase

rn-component

A p

p p(A.H.31)

89

0-component

c1~;~O r )~x~ ( mrVp)- ~c

(A. II. 32)

n-component

(A.I1.33)


Gas Phase

(A. 1I.34),Q.I C'

Liquid Phase

(A.II.35)

l 9

90

d) Variation of G-, with respect to the three coordinate directions:m-direction

on.

_ al rt_) r x(A.I.36)

'; (1 :,\v j -ts~ ( VIN

e-di recti on

Ae (A. 11.37)

n-direction

'r- ri Vn1

9 1 rWr, F

t+ 6

91

(A. II. 38)\&fc~~. '.j{~ (~W~ 0 --rca)

e) Radial Equilibrium Equation:

These equations are deduced utilizing the method of Wennerstrom

(Ref. 56).

Gas Phase

____ v. C. Yr~ Ci -Iv

(A.II.39)

I6 t -,, 4 y ,IL (1P',-

- \. ~V~~ ~I~jCOSC .4 c)

92

$

IIL:iquid Phase

dL~r -Y* Q2/rr --.

r L (4 L

(A. II.40)

- c3SOz-v- )T, r I d. i

It may be pointed out here that in general three-dimensional flow,

it is necessary to set up equilibrium conditions in all of the threedirections (Eqs. A.II.17, A.II.18, A.II.19 and A.II.23, A.II.25, A.11.25).

11.3. Three-dimensional Flow Equations in Cylindrical Coordinates


Gas Phase

r ( - (- T, \,r)

" Jr r ce

L(A

.(1.41

93

Liquid Phase

) (A.H1.42)

b) Momentum Conservation Equations:

Gas Phase

r-component

\d +LO) W

-

0-component

IwL) Y

(A.H1.44)

__, SwF

94

S- compornent

(~~)~wr LO +(w, Z.)~4~

- -P3 -f-Y.t - (A.11. 45)IjLiquid Phase

r-component

1 (A.II.46)

6 -component

(A. 11.47)

95

z-component

( )[r rZ'

- ~ . rr I(A. II.49)

F F

dz cr r r Ls Jo

GL- Vpz 1

S96

P1i

r r (.H.49

n' =~~~~~~~~~~~~~ -S iIilll nnLW,. iam 1 ilIlI q7' --

Liquid Phase

(A. 11.50)

1- do zch- jz7

d) Variation of c3y with respect to the three coordinate directions:

r-direction

The equation is obtained as follows. The 0-component and the z-

component gas phase momentum conservation equations are differentiated

with respect to 0 and z, respectively. Next, the r-component gas phase

momentum conservation equation is solved in teims of ( J -o-v ). These

three equations are then combined with the gas phase mass conservation

equation to obtain the final result.

V + .. I r0 LeJ1

Jr rV/~

97

-- 2.

-O ec tion 1t ro

~ ~L)~~.4G -(A.H.51)

'W \v- + SI,( 4 4t'

+ S64,ZrJ W 7 Lr V/ rW,7SW

A4 F\,.. (5+ ) X. - 7

0-direction

This equation is obtained as follows. The r-component and the z-

component gas phase momentum conservation equations are differentiated

with, respect to r and z, respectively. Next, the O-component gas phase

momentum equation is solved in ters of ( 4-'-v ). These three equations

are then combined with the gas phaie mass conservation equation to obtain

the final result.

98

.0

+ (w) -t)G r-~~t) W~ A

~ d Ka T r e

z rev 4V/v~_ _ _r 1.o

[Wovr % -r 4J-,

99

z-direction

This equation is obtained as follows. The r-component and the e,

compor i.t gas phase momentum conservation equations are differentiated

with respect to r and 0, respectively. Next the z-component Cs phase

momentum conservation equation is solved in terms of ( 1. -Ti. ). These

three equations are then combined with the gas phase mass conservation

equation to obtain the final result.

) t.J..... ( 4/-..1..

I r' ] )o ", 1-# S., . r0,-tF - ! \L <,L j 4

-135-

,,, % .+ .,<. <__... + 1 .[ , _ ,.,.

L i- ,,) -, z- .z ,

j r

y r V). Ir[W~i~~2. (k4~~. J 6 J](A.H1.53)

!(

100 I'I10

/,9. r W

e) Equilibrium Equations:

Gas Phase

These equations are obtained by combining (A.II.16) with each of the

three component gas phase momentum conservation equations.

r-component

- l- a -

101

I__

0-component

(A.II.55)

z-component

-I c z; . (A.II.56)

Liquid Phase

These equations are obtained by combining (A.1I.22) with each of the

three component liquid phase momentum conservation equations.

102

r-coniponent

CT wv p \Alp r \4,!,-

cjyl e4 (A.II.57)

~T 13f -jr Fv ,

e-coniponent

147 i r s, - ~~eV

0-- :) Ls L5 ~ i (A.II.58)

103

z-component

~ .4JTPO~F -1a F2- 2i c Z (A.II.59)

+ )- o, Fp I

11.4. Axisymmetric Flow Equations in Cylindrical Coordinates


Gas Phase

rjr c

(A.II.60)

Liquid Phase

r~~ a-vJ

(A.II.61)

104

b) Momentum Conservation Equations:

Gas Phase

r-component

a -tc3- j -3-L ~ 4rc J2:

(A.II.62)

- ~ -tF, + FYI -

e-component

(A.II.63)

z-coniponent

p \ecr 'r~4? Wr4 j :7 .P

(A.II.64)

Liquid-Phase

r-component

!Yv-n rxw + Wp. Sw r

105

- p t) Y p & r ,. (A.11. 65)

O-component

~ pz F .3 p F - e F l(A .I I .6 6 )

z-comlponent

cr c2j

(A.II.67)


gas Phase

(1 o)g \er + 1 xVCOz ar

3 --

x r ' '(A.II.68)

106

Liquid Phase

(A.II.69)

Jr J2.

d) Variation of cr- with respect to the three coordinate directions:

r-direction

-.-

rr

(A.II.70)

'107

e-direction

Sov cQ(A. 11. 71)

z-direction

W~r ~Wc t - We.rir. W5 2 W

108

-component

IJ r

C - ~ ---- ~ -- = ---+

Z-Comlponent

L z T5 p L (A.II.75)

~~ + F8,.

z ZZ

Liquid Phase

r-component

I ~ + + Q r

110

.... . , l _Ilp'JI. I I I

LY

(A.II.77)

z-component

(A. 11.78)

-_- - : r- 7

II

water ingestion into axial flow compressors · 2011-05-14 · 2.2 compressibility of the mixture 9...

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